# Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra

## TL;DR

This paper derives the Kac determinant formula for the level N representation of the Ding-Iohara-Miki algebra and links its singular vectors to generalized Macdonald functions, advancing understanding of algebraic structures in quantum algebra.

## Contribution

It provides the first explicit formula for the Kac determinant in this algebra and identifies singular vectors with generalized Macdonald functions.

## Key findings

- Kac determinant formula derived for level N representation
- Singular vectors correspond to generalized Macdonald functions
- Enhances understanding of algebraic structures in quantum algebra

## Abstract

In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level $N$ representation of the Ding-Iohara-Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis).